Method and Apparatus for Determining Gas Flux

ABSTRACT

Disclosed embodiments of the present invention provide means to obtain correct gas density and flux measurements using (i) gas analyzer (open-path, or closed-path gas analyzers with short intake tube, for example 1 m long, or any combination of the two); (ii) fast temperature or sensible heat flux measurement device (such as, fine-wire thermocouple, sonic anemometer, or any other device providing fast accurate gas temperature measurements); (iii) fast air water content or latent heat flux measurement device (such as, hygrometer, NDIR analyzer, any other device providing fast accurate gas water content measurements); (iv) vertical wind or sampling device (such as sonic anemometer, scintillometer, or fast solenoid valve, etc.) and (v) algorithms in accordance with the present invention to compute the corrected gas flux, compensated for T-P effects. In case when water factor in T-P effects is negligible, the fast air water content or latent heat flux measurement device (item iii in last paragraph) can be excluded.

BACKGROUND OF THE INVENTION

Embodiments of the present invention relate generally to gas analysisand more particularly to the measurement of gas density to determine gasflux.

When gas density measurements are performed by scanning a singlerotational line or a few discrete lines of analyte with a single-modetunable laser source, the measured signal is temperature and pressuredependent due to the combined effects of Boltzmann populationdistribution of rotational levels and Doppler broadening andtemperature-dependent pressure broadening of individual lines. Thesepressure and temperature effects can also be affected by the presence ofwater and other gases in the sampled air. The combination of all ofthese effects are referred to herein as T-P (temperature-pressure)effects. In addition, if a constant mixing ratio gas is used at arelatively constant pressure, then the measured gas density itselfchanges with temperature and water content due to thermal expansion andwater dilution of the gas per the Ideal Gas Law.

When the temperature or water content of the gas changes, the T-Peffects may lead to a large change in absorption, significantlyaffecting the gas density measurement. In general, the T-P effects arespecific and different for each absorption line of each gas.

For slow measurements of gas density (e.g., measurements taken on theorder of seconds and longer per measurement), the T-P effects can beeasily calibrated out because the mean temperature and water content ofthe gas in the sampling volume can be easily measured. For the case offast measurements of gas density (.e.g., several measurements taken persecond), it is difficult to correct for T-P effects on-the-fly, becauseit would require accurate and precise measurements of gas temperatureand water content integrated over the entire sampling volume. Moreover,the gas temperature and water content measurements would have to berecorded at the same moment when laser absorption due to gas density ismeasured.

Existing gas analyzers, especially for trace gases such as methane,nitrous oxide, isotopes of carbon dioxide and water, etc., areclosed-path sensors requiring long intake tubes and powerful pumps toallow sample gas flow of 30-100 lpm (liters per minute) and more. Thefast temperature changes are attenuated in these long intake tubes soslow temperature measurements can be used, but power consumption of suchsensors systems goes up to 1000 Watts and more, making them difficult touse in remote locations where most of the natural gas exchange processesfor these gases occurs.

BRIEF SUMMARY OF THE INVENTION

Gas analysis in accordance with the present invention allows correctinggas flux for the T-P effects without a need for accurate and precisefast measurements of gas temperature and water content integrated overthe entire sampling volume, and without having to record suchmeasurements at the same exact moment when laser absorption is measured.

Instead, gas analysis per the present invention allows correcting gasflux for the T-P effects using conventional fast measurements of airtemperature and gas water content (if necessary) located away from thegas sampling path, i.e., measurements are not integrated over the gassampling volume, and no need to record temperature and gas water contentat the same precise moment with the taking of laser absorptionmeasurements. In addition, gas analysis according to the presentinvention allows using long term statistics (e.g., 10 minutes to 4hours) to obtain reliable correction for T-P effects over the sameintegration interval as the flux measurements, namely 10 minutes to 4hours. These are generally accepted values for integration time of gasflux calculations. For example, 10 minutes might be appropriate formeasurements taken very close to the ground in a city environment, while4 hours may be appropriate when taking measurement in the middle oflarge uniform desert with little topography. Normal integration timesare 30 minutes to 2 hours.

Gas analysis in accordance with the present invention significantlysimplifies the instrumental requirements for fast gas flux measurementdevices, because instead of making it mandatory to design an instrumentwith fully attenuated (or very well measured) fast gas temperature,pressure, and water content in the entire cell, temperature and gaswater content records (measurements) from external conventional sensors(such as, fine-wire thermocouple, sonic anemometer, or any other deviceproviding fast accurate air temperature measurements, and any fast H₂Oinstrument) positioned near the sampling path of the gas analyzer areadequate. The term “near” refers here to a distance ranging from zero(at or within the sampling path) to several meters from the samplingpath.

As a result, there is no need to design a gas analyzer with fullyattenuated, eliminated, or very well measured fast gas temperature,pressure and water contend in the cell. The teachings of the presentinvention allow designing relatively simple devices (open-path, orclosed-path gas analyzers with short intake tube, for example 1 m long,or any combination of the two) having power demands 50-100 times belowconventional gas analyzers, yet with similar accuracy of gas fluxmeasurements.

Aspects of the present invention include accurately relating T-P effectscaused by a fast change in temperature to the thermal expansion effect(as per Ideal Gas Law) caused by the same fast change in temperature,and accurately relating T-P effects caused by a fast change in gas watercontent to the water dilution effect (as per Ideal Gas Law) caused bythe same fast change in gas water content. Thus, well-quantifiedprocesses of thermal expansion and water dilution are used for thecorrection of temperature expansion, water dilution and T-P effects.

The capabilities of gas analysis according to the present invention areextremely important and useful in areas of research such as gas exchangestudies, climate studies, and atmospheric experimental research becauseit is possible, using commercially available lasers, to design low-powergas analyzers for Eddy Covariance, Eddy Accumulation, airborne andmarine methods and for other “fast” methods of measuring gas exchange,requiring accurate and precise gas density measurements several or moretimes a second. With the ability to correct for T-P effects due to highfrequency temperature and air water content fluctuations, open-pathsensors and closed-path sensors with short wide intake tubes can beprovided to the scientific and gas monitoring communities, reducingpower consumption of the scientific and monitoring systems from 1000Watts to 10-20 Watts. Such sensors can be placed in remote locations ofinterest, powered by solar panels, or can be used as portable hand-heldsensors for gas density measurements.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a typical embodiment of a gas analysis system inaccordance with the present invention.

FIG. 2 shows a system block diagram of a typical embodiment of a datahandling system in accordance with the present invention.

FIG. 3 shows the processing for determining flux values corrected fortemperature and pressure in accordance with one embodiment of thepresent invention.

FIG. 4 shows the processing for determining flux values corrected fortemperature and pressure in accordance with another embodiment of thepresent invention.

FIGS. 5A and 5B show calibration curves for a gas analyzer.

FIG. 6 represents results from actual test data, comparing conventionaldeterminations of flux with determination of flux per the presentinvention.

DETAILED DESCRIPTION OF THE INVENTION

Techniques of the present invention provide means to obtain correct gasdensity and flux measurements using (i) a gas analyzer (open-path, orclosed-path gas analyzers with short intake tube, for example 1 m long,or any combination of the two); (ii) a fast temperature or sensible heatflux measurement device (such as, fine-wire thermocouple, sonicanemometer, or any other device providing fast accurate gas temperaturemeasurements); (iii) a fast air water content or latent heat fluxmeasurement device (such as, hygrometer, NDIR analyzer, any other deviceproviding fast accurate gas water content measurements); (iv) a verticalwind or sampling device (such as sonic anemometer, scintillometer, orfast solenoid valve, etc.) and (v) algorithms in accordance with thepresent invention to compute a corrected gas flux that compensated forT-P effects. In a situation when the water factor in T-P effects isnegligible, the fast air water content or latent heat flux measurementdevice (item iii above) can be excluded.

Teachings of the present invention are adaptable for any open-path orclosed-path (with short intake tube, for example, 1 m long) gas analyzerthat provide fast response measurements of gas density (several timesper second). Embodiments of the present invention include the use of atemperature or sensible heat flux measurement device (such as afine-wire thermocouple, sonic anemometer, scintillometer, etc.)providing fast accurate gas temperature or sensible heat fluxmeasurements. In addition, when the water factor in T-P effects is notnegligible, fast air water content or latent heat flux measurementdevice (described previously) or an estimate of mean latent heat flux isalso factored in.

For the specific case of Eddy Covariance measurements of gas flux, a gasanalyzer per the present invention can use fast measurements of verticalwind speed, and is readily adapted to use any wind speed measurementsdevice (such as, sonic anemometer, etc.) that provides fast accuratemeasurements of vertical wind speed.

Generally, embodiments of the present invention comprise the following:

-   -   1. Calibrate gas analyzer versus temperature or use HITRAN (High        resolution TRANsmission Molecular Absorption Database), and        establish a T-P response surface describing T-P effects.    -   2. Install a fast temperature or sensible heat flux measurement        device near the gas analyzer. Record fast temperature or        sensible heat flux alongside the fast gas density.        -   a. In case the water factor in T-P effects is not            negligible, install a fast air water content or latent heat            flux measurement device. Record fast air water content or            latent heat flux alongside the fast gas density.        -   b. For the case of Eddy Covariance gas flux measurements,            install the anemometer device near gas analyzer, and record            vertical wind speed alongside fast gas density.    -   3. Compute, measure, or estimate sensible heat flux by any        conventionally known method. In case when water factor in T-P        effects is not negligible, compute, measure, or estimate latent        heat.    -   4. For the case of Eddy Covariance gas flux measurements,        combine and align recorded time series of vertical wind speed        and gas density measurements on a longer-term basis (minutes to        hours), and compute raw uncorrected gas flux.    -   5. Compute gas flux corrected for T-P effects on spectral        absorption. Frequency of data collection and averaging period        (minutes to hours) are determined by the specific approach and        purpose of the gas measurements.

A gas analyzer in accordance with the present invention provides gasflux that is corrected for T-P effects without the need for makingmeasurements of fast gas temperature or air water content integratedover the gas sampling volume. This enables the use of low-poweropen-path gas analyzers as well as the use of reduced-power closed-pathanalyzers with short intake tubes. It is believed that no such analyzersrelying on a single line, or narrow absorption range, are currentlyavailable.

First, a discussion will be given of algorithms and procedures accordingto the present invention used to obtain gas flux measurements that arecorrected for the temperature and pressure conditions that existedduring the collection of the measurement data. The discussion will thenturn to illustrative embodiments of gas analysis in accordance with thepresent invention.

The discussion that follows explains the derivation of an algorithm forEddy Covariance gas flux. In particular, the Webb-Pearman-Leuning (WPL)density term will be examined and modified in accordance with thepresent invention to account for T-P effects.

Derivation of Algorithm for Eddy Covariance Gas Flux Measurements

A. General Form of Propagation of T-P Effects of a Single-Line or NarrowRange Laser Measurement into Eddy Covariance Flux Measurements

Let us define χ (chi) as the ratio of actual to measured gas densitiesas affected by T-P effects only, namely χ=J(T, P, etc.), excludingtemperature-related gas expansion and water-related gas dilutioneffects. The function χ is also referred to variously as the instrumentmeasurement response, instrument response, response function,measurement response function, and so on. In the most general form, χcould also include the effects of pressure broadening by water vapor andother possible air constituents, other spectroscopic effects andinstrument-specific and method-specific response functions. χ could bewritten then in general form as follows:

ρ_(c)=ρ_(cm)χ  (1)

where ρp_(cm) is measured gas density that is not corrected for T-Peffects, and ρ_(c) is gas density that is corrected for T-P effects.Using Reynolds decomposition, each term can be written into following:

ρ_(c)=ρ_(c)′+ ρ_(c)   (2)

ρ_(cm)=ρ_(cm)′+ ρ_(cm)   (3)

χ=χ′+ χ  (4)

where mean (average) quantities are indicated by the over-bar notation,the deviation of instantaneous quantity (i.e., the measured quantity)from the mean is indicated by a prime (′), and instantaneous quantity isindicated by the absence of the over-bar or prime symbols.

Combining Equations 1-4 leads to:

ρ_(c)′+ ρ_(c) =ρ_(cm)′χ+ ρ_(cm) χ  (5 a)

ρ_(c)′+ ρ_(c) =ρ_(cm)′χ′+ρ_(cm)′ χ+ ρ_(cm) χ′+ ρ_(cm)χ  (5b)

and computing co-variances for flux computation results in:

w′ρ _(c)′+ w′{overscore (ρ_(c))}= w′ρ _(cm)′χ′+ w′ρ _(cm)′χ+ ρ_(cm)w′χ′+ w′{overscore (ρ_(cm)χ)}  (6)

The terms where instantaneous quantity is correlated with mean quantityare cancelled, such that:

w′ρ _(c)′= w′ρ _(cm)′χ′+ w′ρ _(cm)′χ+ ρ_(cm) w′χ′  (7)

and after re-arranging the order in Equation 7, the following equationis constructed:

w′ρ _(c)′= w′ρ _(cm)′χ′+ w′χ′ρ_(cm) + w′ρ _(cm)′χ′  (8)

The Equation 8 is a general form of propagation of T-P effects of asingle-line or narrow range laser measurement into the Eddy Covarianceflux measurements.

The actual flux co-variance (left term, also referred to herein as “rawflux”) is equal to measured co-variance multiplied by the mean χ (meanT-P effects over an averaging period; first term on the right), pluscovariance between w′ and χ′ (instantaneous T-P effects; second term onthe right), and co-variance of the three prime quantities (last term onthe right). The last term may or may not be negligible depending on thespecific form of the function χ=

(T, P).

B. General Form of Full Equation for Gas Flux Computation, IncludingEffect of Water Dilution, Thermal Expansion, and T-P Effects.

The general form of propagation of T-P effects of a single-line ornarrow range laser measurement into the Eddy Covariance fluxmeasurements, Equation 8, can now be incorporated into full fluxequation including both water dilution and thermal expansion effects (asper Ideal Gas Law) and T-P effects.

For Eddy Covariance Gas flux measurements, the Webb-Pearman-Leuning(WPL) density formulation can be written in the following form for theflux of a non-reactive gas:

$\begin{matrix}{F_{c} = {\overset{\_}{w^{\prime}\rho_{c}^{\prime}} + {\mu \frac{E}{\overset{\_}{\rho_{d}}}\frac{\overset{\_}{\rho_{c}}}{1 + {\mu \frac{\overset{\_}{\rho_{v}}}{\overset{\_}{\rho_{d}}}}}} + {\frac{S}{\overset{\_}{\rho}C_{p}}\frac{\overset{\_}{\rho_{c}}}{\overset{\_}{T}}}}} & (9)\end{matrix}$

with water vapor flux E:

$\begin{matrix}{E = {\left( {1 + {\mu \frac{\overset{\_}{\rho_{v}}}{\overset{\_}{\rho_{d}}}}} \right)\left( {E_{0} + {\frac{S}{\overset{\_}{\rho}C_{p}}\frac{\overset{\_}{\rho_{v}}}{\overset{\_}{T}}}} \right)}} & (10)\end{matrix}$

where F_(c) is WPL-corrected gas flux; E is WPL-corrected H₂O flux;w′ρ_(c)′ is initial CH₄ flux, not corrected for WPL; E_(o) is H₂O flux,not corrected for WPL; μ is ratio of molar masses of air to water(μ=1.6077); ρ_(d) is mean dry air density; ρ_(v) is mean water vapordensity; ρ is mean total air mass density; S is sensible heat flux (Wm⁻²); C_(p) is specific heat of air; T is gas temperature; w is verticalwind speed.

It is important to mention here, that the Webb-Pearman-Leuning densityformulation (WPL) assumes that gas densities have been measuredcorrectly, except for water dilution and thermal expansion effects asper Ideal Gas Law. So, in our terminology defined in Equations 1-3,ρ_(c) should be used in the WPL formulation in Equation 9, and notρ_(cm).

As follows from the Equation 1, the ρ_(c) = ρ_(cm)χ. Then, combining itwith Equations 8 and 9 yields:

$\begin{matrix}{F_{c} = {\overset{\_}{\underset{\underset{1\; {st}\mspace{14mu} {member}}{}}{w^{\prime}\rho_{cm}^{\prime}\chi}} + \underset{\underset{2\; {nd}\mspace{14mu} {member}}{}}{\mu \frac{E}{\overset{\_}{\rho_{d}}}\frac{\overset{\_}{\rho_{cm}\chi}}{1 + {\mu \frac{\overset{\_}{\rho_{v}}}{\overset{\_}{\rho_{d}}}}}} + \underset{\underset{3\; {rd}\mspace{14mu} {member}}{}}{\frac{S}{\overset{\_}{\rho}C_{p}}\frac{\overset{\_}{\rho_{cm}\chi}}{\overset{\_}{T}}} + \overset{\_}{\underset{\underset{4\; {th}\mspace{14mu} {member}}{}}{w^{\prime}\chi^{\prime}\rho_{cm}}} + \overset{\_}{\underset{\underset{5\; {th}\mspace{14mu} {member}}{}}{w^{\prime}\rho_{cm}^{\prime}\chi^{\prime}}}}} & (11)\end{matrix}$

The Equation 11 is a full equation for gas flux computation, includingeffect of water dilution, thermal expansion, and T-P effects.

The first member on the right side of Equation 11 is related to rawuncorrected flux in the original WPL formulation modified due to meanT-P effects. The term w′ρ_(cm)′ in the first member is sometimesreferred to in the literature as the uncorrected flux in the WPLformulation, and is computed from the measured data. This term isreferred to herein as “raw” flux. The term is modified according to thepresent invention, as expressed in Equation 11, to account for T-Peffects.

Similarly, the second member is related to effect of water dilution inthe original WPL formulation modified due to mean T-P effects. The term

$\mu \frac{E}{\overset{\_}{\rho_{d}}}\frac{\overset{\_}{\rho_{cm}}}{1 + {\mu \frac{\overset{\_}{\rho_{v}}}{\overset{\_}{\rho_{d}}}}}$

in the second member is understood by those of ordinary skill torepresent, with respect to the WPL formulation, the effect due to waterdilution of the target gas on the measured flux of the target gas, andis a term that is computed from the measured data. This term, referredto herein generally as water dilution, is modified according to thepresent invention, as expressed in Equation 11, to account for T-Peffects.

The third member is related to the effect of thermal expansion in theoriginal WPL formulation modified due to mean T-P effects. The term

$\frac{S}{\overset{\_}{\rho}C_{p}}\frac{\overset{\_}{\rho_{cm}}}{\overset{\_}{T}}$

in the third member is understood by those of ordinary skill torepresent, with respect to the WPL formulation, a thermal expansioneffect on the measured flux of the target gas, and is generally computedfrom the measured data. This term, referred to herein generally asthermal expansion, is modified according to the present invention, asexpressed in Equation 11, to account for T-P effects.

The fourth and fifth members on the right side of the equation 11 areentirely new, and do not have equivalent terms in the original WPLformulation. These members describe instantaneous T-P effects. Thelatter of the two, the fifth member, may or may not be negligibledepending on the form of function χ=

(T, P) in the specific instrument and technique for a specific laser andgas specie.

To better understand the physical meaning of Equation 11, the equationcan be re-arranged as follows:

$\begin{matrix}{F_{c} = {{\left( {\overset{\_}{w^{\prime}\rho_{cm}^{\prime}} + {\mu \frac{E}{\overset{\_}{\rho_{d}}}\frac{\overset{\_}{\rho_{cm}}}{1 + {\mu \frac{\overset{\_}{\rho_{v}}}{\overset{\_}{\rho_{d}}}}}} + {\frac{S}{\overset{\_}{\rho}C_{p}}\frac{\overset{\_}{\rho_{cm}}}{\overset{\_}{T}}}} \right)\overset{\_}{\chi}} + \overset{\_}{w^{\prime}\chi^{\prime}\rho_{cm}} + \overset{\_}{w^{\prime}\rho_{cm}^{\prime}\chi^{\prime}}}} & (12)\end{matrix}$

So, T-P effects propagate into flux calculation as a χ multiplier to thetraditionally computed fully corrected flux (i.e., all the membersinside the parenthesis in Equation 12). This multiplier is compensatedfor mean T-P effects. Two additional terms w′χ′ρ_(cm) and w′ρ_(cm)′χ′compensate for instantaneous T-P effects.

The measured gas density ρ_(cm) is a measurement that comes from the gasanalyzer. The temperature and pressure for χ are obtained from the fasttemperature and pressure measurement devices (either external orincorporated into the gas analyzer). The form of the χ function can beobtained from a calibration curve of the gas analyzer by performing acalibration of the instrument and/or using data from the HITRAN databaseor other such similar database. In general, any of a number of knowntechniques for calibrating an instrument can be used. Vertical windspeed typically is obtained from an anemometer. The other terms arewell-known physical constants and standard atmospheric parameters.

1. Specific Derivation for the Case of the LI-7700 Methane Gas Analyzer

The LI-7700 is a methane gas analyzer developed, manufactured, and soldby the Assignee of the present invention. The LI-7700 has a measurementresponse (also referred to as a calibration curve or response curve orresponse function) that is dependent on temperature and pressure. ThisT-P dependence arises from a number of effects: changes in the Boltzmannpopulation distribution of the rotational levels, Doppler and pressurebroadening of individual lines. All of these effects have beencalculated for the following conditions: 50 to 110 kPa (pressure rangein kilopascals) and from 233K to 323K (temperature range in kelvin).These calculated absorption profiles were then run through themodulation/demodulation algorithm and the predicted responses werecollected into a table. The validity of that correction table has beenconfirmed in controlled laboratory studies.

a) LI-7700 Instrument Calibration

The LI-7700 generates a 2F demodulated waveform of the absorptionprofile at some temperature and pressure. The first step in calibrationis to use a zero gas to subtract any offsets to this waveform. A zerogas is flowed through the sample path. The raw data, α _(zero), isrecorded for 20s. This value must be collected before spanning.

When spanning the instrument, a tank of methane/air balance is used. Themole fraction of this tank is known by the user. This gas is flowedthrough the LI-7700 sample path and the value of the mole fraction isentered into the software. When the calibration button is pressed, datafor temperature, pressure and raw absorption is collected and used tocalculate as span value. First, the gas density is calculated asfollows:

$\begin{matrix}{\rho_{span} = \frac{x_{cal}R{\overset{\_}{P}}_{span}}{{\overset{\_}{T}}_{span}}} & (13)\end{matrix}$

where ρ_(span) is the calculated density of the calibration gas, x_(cal)is the user entered mole fraction value, R is the universal gasconstant, and P _(span) and T _(span) are the mean pressure andtemperature calculated over 20 seconds. This density value is used tocalculate a span constant:

$\begin{matrix}{C_{span} = \frac{\rho_{span}}{\left( {{\overset{\_}{\alpha}}_{span} - {\overset{\_}{\alpha}}_{zero}} \right){f\left( {{\overset{\_}{T}}_{span},{\overset{\_}{\rho}}_{span}} \right)}}} & (14)\end{matrix}$

where C_(span) is the span constant, α _(span) is the mean of the rawabsorption calculated over 20 seconds, α _(zero) is the mean of the rawabsorption calculated from the zeroing procedure, and f( T _(span), ρ_(span)) is a general function describing the instrument measurementresponse. This is given by:

$\begin{matrix}{{f\left( {\overset{\_}{T},\overset{\_}{P}} \right)} = \frac{\rho_{c}}{\rho_{cm}}} & (15)\end{matrix}$

where ρ_(c) is the actual gas density and ρ_(cm) is the measured gasdensity. Measured gas density is reported in the analyzer as follows:

ρ_(cm) =C _(span)(α− α _(zero))  (16)

where α is the instantaneous raw absorption measured on the analyzer.

b) f(T,P) Propagation Through EC Flux Measurements

The response correction function and measured and actual density can bewritten as the following:

ρ_(cm)= ρ _(cm)+ρ_(cm)′  (17)

ρ_(c)= ρ _(c)+ρ_(c)′  (18)

f(T,P)=f( T, P )+f′( T, P )T′+H.O.T.  (19)

We can assume that the higher order terms of Taylor's expansion of fwill be negligible. The function f indicates a derivative off and not aperturbation from the mean of f. For given instrument, water factor inT-P effects is neglected in Equation 19 in this example because of itsexperimentally confirmed low importance.

From equation 3, 5, and 7, the actual density is:

ρ_(c)=( ρ _(cm)+ρ_(cm)′)(f( T, P )+f′( T, P )T′)  (20)

Expanded form:

ρ_(c)= ρ _(cm) f( T, P )+ρ_(cm) ′f( T, P )+ ρ _(cm) f′( T, P )T′+ρ′_(cm) f′( T, P )T′  (21)

Substituting 15 and 21 into 18 and solving for ρ_(c)′ gives:

ρ_(c)′= ρ _(cm) f( T, P )+ρ_(cm) ′f( T, P )+ ρ _(cm) f′( T, P )T′+ρ_(cm) ′f′( T, P )T′− ρ _(cm) f( T, P )  (22)

ρ_(c)′=ρ_(cm) ′f( T , P )+ ρ _(cm) f′( T , P )T′+ρ _(cm) ′f′( T ,P)T′  (23)

Calculating co-variances for fluxes gives:

$\begin{matrix}{\overset{\_}{w^{\prime}\rho_{c}^{\prime}} = {\overset{\_}{w^{\prime}\rho_{cm}^{\prime}{f\left( {\overset{\_}{T},\overset{\_}{P}} \right)}} + \overset{\_}{w^{\prime}{\overset{\_}{\rho}}_{cm}{f^{\prime}\left( {\overset{\_}{T},\overset{\_}{P}} \right)}T^{\prime}} + \overset{\_}{w^{\prime}\rho_{cm}^{\prime}{f\left( {\overset{\_}{T},\overset{\_}{P}} \right)}T^{\prime}}}} & (24)\end{matrix}$

For now, we will assume the 3rd term of 24 is negligible, giving:

w′ρ′ _(c)= w′ρ′_(cm) f( T , P )+ ρ _(cm) w′T′f′( T , P )  (25)

Equation 25 can now be substituted into the WPL formulation for fluxshown below.

$\begin{matrix}{\mspace{79mu} {F_{c} = {\overset{\_}{w^{\prime}\rho_{c}^{\prime}} + {\mu \frac{E}{{\overset{\_}{\rho}}_{d}}\frac{{\overset{\_}{\rho}}_{c}}{1 + {\mu \frac{{\overset{\_}{\rho}}_{v}}{{\overset{\_}{\rho}}_{d}}}}} + {\frac{S}{\overset{\_}{\rho}C_{p}}\frac{{\overset{\_}{\rho}}_{c}}{\overset{\_}{T}}}}}} & (26) \\{F_{c} = {{\overset{\_}{w^{\prime}\rho_{cm}^{\prime}}{f\left( {\overset{\_}{T},\overset{\_}{P}} \right)}} + {{\overset{\_}{\rho}}_{cm}\overset{\_}{w^{\prime}T^{\prime}}{f^{\prime}\left( {\overset{\_}{T},\overset{\_}{P}} \right)}} + {\mu \frac{E}{{\overset{\_}{\rho}}_{d}}\frac{{\overset{\_}{\rho}}_{cm}{f\left( {\overset{\_}{T},\overset{\_}{P}} \right)}}{1 + {\mu \frac{{\overset{\_}{\rho}}_{v}}{{\overset{\_}{\rho}}_{d}}}}} + {\frac{S}{\overset{\_}{\rho}C_{p}}\frac{{\overset{\_}{\rho}}_{cm}{f\left( {\overset{\_}{T},\overset{\_}{P}} \right)}}{\overset{\_}{T}}}}} & (27)\end{matrix}$

Given,

$\begin{matrix}{S = {{\overset{\_}{\rho}\; C_{p}\overset{\_}{w^{\prime}T^{\prime}}\mspace{14mu} {or}\mspace{14mu} \frac{w^{\prime}T^{\prime}}{\overset{\_}{T}}} = \frac{S}{\overset{\_}{\rho}\; C_{p}\overset{\_}{T}}}} & (28)\end{matrix}$

then Equation 27 can be arranged as:

$\begin{matrix}{F_{c} = {{w^{\prime}\rho_{cm}^{\prime}{f\left( {\overset{\_}{T},\overset{\_}{P}} \right)}} + {{\overset{\_}{\rho}}_{cm}\frac{S}{\overset{\_}{\rho}\; C_{p}\overset{\_}{T}}\overset{\_}{T}{f^{\prime}\left( {\overset{\_}{T},\overset{\_}{P}} \right)}} + {\mu \frac{E}{\overset{\_}{\rho_{d}}}\frac{{\overset{\_}{\rho}}_{cm}{f\left( {\overset{\_}{T},\overset{\_}{P}} \right)}}{1 + {\mu \frac{{\overset{\_}{\rho}}_{v}}{{\overset{\_}{\rho}}_{d}}}}} + {\frac{S}{\overset{\_}{\rho}\; C_{p}}\frac{{\overset{\_}{\rho}}_{cm}{f\left( {\overset{\_}{T},\overset{\_}{P}} \right)}}{\overset{\_}{T}}}}} & (29) \\{F_{c} = {{\overset{\_}{w^{\prime}\rho_{cm}^{\prime}}{f\left( {\overset{\_}{T},\overset{\_}{P}} \right)}} + {\mu \frac{E}{{\overset{\_}{\rho}}_{d}}\frac{{\overset{\_}{\rho}}_{cm}{f\left( {\overset{\_}{T},\overset{\_}{P}} \right)}}{1 + {\mu \frac{{\overset{\_}{\rho}}_{v}}{{\overset{\_}{\rho}}_{d}}}}} + {\frac{S}{\overset{\_}{\rho}\; C_{p}}{\frac{{\overset{\_}{\rho}}_{cm}}{\overset{\_}{T}}\left\lbrack {{f\left( {\overset{\_}{T},\overset{\_}{P}} \right)} + {\overset{\_}{T}{f^{\prime}\left( {\overset{\_}{T},\overset{\_}{P}} \right)}}} \right\rbrack}}}} & (30)\end{matrix}$

This form of the flux equation allows for the use of a general functionfor the measure response of the LI-7700 instrument to be applied to halfhour data rather than on instantaneous data. The derivative off can becomputed very easily from curve fits to the T-P curve or numericallyfrom table values.

Validation of the Proposed Method Using LI-7700 Methane Analyzer andField Data

The general equations 8 and 11 and instrument-specific equations 24 and30 were validated on the example of LI-7700 fast methane analyzer (2009version of instrument and methodology of scanning methane line). Theform of χ in this example is shown in a normalized form in FIGS. 5A and5B using linear (FIG. 5A) and exponential (FIG. 5B) fits. For a giveninstrument, water factor in T-P effects is neglected in this examplebecause of its experimentally confirmed low importance.

Using linear fit in this specific example, the χ can be approximated asfollows:

$\begin{matrix}{\chi \approx {{0.32\frac{T}{\overset{\_}{T}}} + 0.67}} & (31) \\{{\overset{\_}{\chi} \approx {{0.32\frac{\overset{\_}{T}}{\overset{\_}{T}}} + 0.67}} = 0.99} & (32) \\{{\chi^{\prime} \approx \left( {{0.32\frac{T}{\overset{\_}{T}}} + 0.67} \right)^{\prime}} = {\frac{0.32}{\overset{\_}{T}}T^{\prime}}} & (33)\end{matrix}$

Combining Equations 11 and 32-33 yields:

$\begin{matrix}{F_{c} = {{0.99\overset{\_}{w^{\prime}\rho_{cm}^{\prime}}} + {0.99\mu \frac{E}{\overset{\_}{\rho_{d}}}\frac{\overset{\_}{\rho_{cm}}}{1 + {\mu \frac{\overset{\_}{\rho_{v}}}{\overset{\_}{\rho_{d}}}}}} + {0.99\frac{S}{\overset{\_}{\rho \;}C_{p}}\frac{\overset{\_}{\rho_{cm}}}{\overset{\_}{T}}} + {\frac{0.32}{\overset{\_}{T}}\overset{\_}{w^{\prime}T^{\prime}\rho_{cm}}} + {\frac{0.32}{\overset{\_}{T}}\overset{\_}{w^{\prime}\rho_{cm}^{\prime}T^{\prime}}}}} & (34)\end{matrix}$

The right most member of the equation 34 becomes negligible, because theproduct is several orders of magnitude smaller than other members,yielding:

$\begin{matrix}{F_{c} = {{0.99\; \overset{\_}{w^{\prime}\rho_{cm}^{\prime}}} + {0.99\mu \frac{E}{\overset{\_}{\rho_{d}}}\frac{\overset{\_}{\rho_{cm}}}{1 + {\mu \frac{\overset{\_}{\rho_{v}}}{\overset{\_}{\rho_{d}}}}}} + {0.99\frac{S}{\overset{\_}{\rho \;}C_{p}}\frac{\overset{\_}{\rho_{cm}}}{\overset{\_}{T}}} + {\frac{0.32}{\overset{\_}{T}}\overset{\_}{w^{\prime}T^{\prime}\rho_{cm}}}}} & (35)\end{matrix}$

Finally, using standard equation for sensible heat flux, S= ρC_(p)w′T_(a)′, the equation 35 becomes:

$\begin{matrix}{F_{c} = {{0.99\overset{\_}{w^{\prime}\rho_{cm}^{\prime}}} + {0.99\mu \frac{E}{\overset{\_}{\rho_{d}}}\frac{\overset{\_}{\rho_{cm}}}{1 + {\mu \frac{\overset{\_}{\rho_{v}}}{\overset{\_}{\rho_{d}}}}}} + {1.31\frac{S}{\overset{\_}{\rho}\; C_{p}}\frac{\overset{\_}{\rho_{cm}}}{\overset{\_}{T}}}}} & (36)\end{matrix}$

The Equation 35 is a specific sub-case of Equation 30 for givenatmospheric pressure and given calibration temperature and pressure,while Equations 30 is a specific sub-case of Equation 11 for giveninstrument. It describes propagation of both T-P effects and densityeffects (per Ideal Gas Law) into the Eddy Covariance flux calculationsusing LI-7700 gas analyzer.

Using exponential fit instead of linear fit (Eqs. 31-33) in thisspecific example, the function χ can be approximated as follows:

$\begin{matrix}{\mspace{79mu} {\chi \approx {0.712\; {\exp \left( {0.323\frac{T}{\overset{\_}{T}}} \right)}}}} & (37) \\{\mspace{79mu} {{\overset{\_}{\chi} \approx {0.712\; {\exp \left( {0.323\frac{\overset{\_}{T}}{\overset{\_}{T}}} \right)}}} = 0.98}} & (38) \\{{\chi^{\prime} \approx {0.712*\frac{0.323}{\overset{\_}{T}}*T^{\prime}{\exp \left( {0.323\frac{T}{\overset{\_}{T}}} \right)}}} = {{{\frac{0.23}{\overset{\_}{T}}T^{\prime}{\exp \left( {0.323\frac{T}{\overset{\_}{T}}} \right)}} \approx {\frac{0.23}{\overset{\_}{T}}T^{\prime}{\exp (0.323)}}} = {\frac{0.32}{\overset{\_}{T}}T^{\prime}}}} & (39)\end{matrix}$

Either linear or exponential fit yield substantially same results, withidentical χ′ and similar χ-means. The 1% difference in χ-means is likelydue to an imperfections of linear fit over the wide range oftemperatures (−40 to +50° C.).

Since forms of Equations 31 or 37 vary in LI-7700 (2009 version ofinstrument and methodology of scanning methane line) with actual andcalibration gas temperatures and pressures, the fitting is not thepreferable approach in this case. High-resolution look-up table wouldprovide a more accurate value of χ for every specific time and set ofconditions.

To validate Equations 11, 30 and 36, it is convenient to compare methanefluxes from the non-producing field site with and without the correctionfor T-P effect. FIG. 6 illustrates such a test, showing an ensemble ofaveraged flux data from two weeks in the Summer of 2009, over anagricultural field in Mead, Nebr.

The experimental site had a long history of chamber measurements of verysmall CH₄ fluxes (−0.1 to 0.1 mg/m² per hour, year round). As seen inFIG. 6, when the proposed method is not used to correct the measuredflux, the measurements are incorrect, exceeding the expected range offluxes by up to 10 times. When the proposed method is used, themeasurements become not significantly different (standard error on eachpoint is about 0.19 mg/m² per hour) from the numbers measured bychambers. This is strong evidence that the method in accordance with thepresent invention works correctly.

The discussion will now turn to a description of FIG. 1. The figureillustrates several elements of an example of a gas analysis systemaccording to the present invention. Those of ordinary skill willappreciate that the system shown in FIG. 1 may include additionalelements not shown in the figure. However, it is not necessary that allof these generally conventional elements be shown in order to disclosean illustrative embodiment for practicing the present invention.

The illustrative gas analysis system shown in FIG. 1 can be viewed ascomprising two sub-systems: a measurement sub-system 122 and a data andcontrol sub-system 124. A data communication component 126 allows dataand control information to be communicated between the two sub-systems122, 124. The data communication component 126 can use any of a numberof conventionally known data communication techniques. The datacommunication component 126 can be wireless. This would be preferable ifthe measurement sub-system is at a location that is remote, or otherwisenot readily accessible, from the personnel (scientists, engineers, etc)who are collecting the data. Examples of wireless configurations,include for example, a radio frequency communication link, an opticalcommunication link, and so on. The data communication component 126 cancomprise hardwired (“wired”) connections such as ethernet cabling,RS-232 communication using a modem, high speed connection (USB,firewire, etc.) to a data logging device, or any other suitable wiredconfiguration. Finally, the data communication component 126 can be somecombination of wired and wireless communication, depending the specificconfigurations of the elements of the gas analyzer system.

FIG. 1 shows that the data communication component 126 includes aplurality of data/communication lines that connect each instrument inthe measurement sub-system 122 to the control sub-system 124. Theillustrated data/communication lines can be physical wires/cabling, or aform of wireless communication link (e.g., radio), or some combinationof both. In the configuration shown, each instrument in the measurementsub-system 122 communicates with the data and control sub-system 124.Alternatively, two or more of the instruments can be connected togetherin serial fashion (e.g., daisy chained) and communications with the dataand control sub-system 122 can occur over a single communication line.

The measurement sub-system 122 includes a gas analyzer 102, a wind speedmeasuring device 104, a water vapor analyzer 106, and a temperaturesensor 108. The gas analyzer 102 is any conventionally known analyzersuitable for measuring the density of a target gas; i.e., the gas ofinterest that is to be analyzed. For example, methane (CH₄) is acommonly measured gas and methane analyzers for measuring methanedensity are commercially available, such as the LI-7700 Open Path CH₄Analyzer, designed, manufactured, and sold by the assignee of thepresent invention. Generally, absorption based gas analyzers useabsorption of light from either (i) a broadband non-dispersive infrared(NDIR) source equipped with suitable optical filter or (ii) a narrowbandlaser source to measure the density of the target gas of interest. Thelight is selectively absorbed by the gas as it crosses the light pathbetween the light source and a detector in a region called the samplingvolume (also variously referred to as “sample volume,” “sampling path,”and so on).” The gas analyzer 102 outputs gas density measurement databased on the measured absorption characteristics. Two categories of gasanalyzers are conventionally known which are defined by the nature ofthe sampling volume. An “open path” type gas analyzer is one in whichthe sampling volume and the optical path are exposed to the environmentcontaining the gas to be analyzed. A “closed path” gas analyzer is onein which the sampling volume is enclosed in a tube (in which case thesampling volume can be referred to as the sample cell) and the opticalpath lies within the tube, and the gas to be measured is passed withinthe tube. In accordance with the present invention, the gas analyzer 102can be either an open path analyzer or a closed path analyzer or acombination of the two. For example, U.S. Pat. Nos. 6,317,212 and6,369,387, each of which is hereby incorporated by reference in itsentirety, disclose various features of open and closed path gasanalyzers.

The wind speed measuring device 104 produces a measure of the speed ofthe moving air in the vicinity of the gas analyzer 102 and outputscorresponding wind speed measurement data. More specifically, the windspeed measurement in accordance with the present invention is verticalwind speed. An instrument commonly used to measure wind speed is knownas a sonic anemometer. This instrument is commonly used with open pathgas analyzers. There are several types of sonic anemometers, ranging incomplexity. The most basic models of sonic anemometers measure the windspeed, while the more complex ones can measure wind speed, winddirection, and wind pressure. It will be appreciated of course thatother wind speed measurement devices and techniques can be used.

The water vapor analyzer 106 provides a measure of the water content ofthe target gas and produces a series of water vapor measurement data. Asexplained above, the amount of light absorption by the target gas isaffected by temperature and pressure. Water vapor analyzers arecommercially available, and any such commercially available analyzer canbe used with embodiments of the present invention. An example of asuitable water vapor analyzer 106 is the LI-7200 CO₂/H₂O Analyzer,designed, manufactured, and sold by the assignee of the presentinvention.

In certain environments, the water content can be significant enough toconsiderably affect the absorption lineshape of the target gas and theresulting density measurement. Dilution by water vapor causes an actualphysical change in partial pressure and a change in actual density whencompared to dry. In addition, water vapor affects absorption by linebroadening which consequently affects the resulting density measurement.Under such conditions, more accurate results will be achieved if thewater content is measured and factored into the computations. However,if the environment where the target gas is being analyzed issufficiently dry, the water content may not have any significant affecton density measurements of the target gas. In that case, the cost andcomplexity of coordinating gas density measurements with water vapormeasurements can be dispensed with and the water vapor analyzer 106would not be required to correct for T-P effects.

The temperature sensor 108 is used to measure ambient temperature in theproximity of the gas density measurements. Typical devices for thetemperature sensor 108 include a fine-wire thermocouple, a sonicanemometer, and in general any device that can provide fast gastemperature measurements. In accordance with the present invention, thetemperature sensor 108 can be positioned in proximity to the samplingvolume of the gas analyzer 102, or alternatively within the samplingvolume.

The data and control sub-system 124 includes a suitable data processingcomponent 112, data storage 114, and a suitable user interface 116.Those of ordinary skill in the art will appreciate that the data andcontrol sub-system 124 may include many more components than those shownin FIG. 1. However, it is not necessary that all of these generallyconventional components be shown in order to disclose an illustrativeembodiment for practicing the present invention.

Typical devices that can serve as the data storage 114 includetraditional disk storage devices such as hard disk drives, floppy diskdrives, writable CD-ROMs, other removable storage formats, and the like.Data storage 114 can also include flash memory devices such as flashdrives, or other similar static storage devices. Data storage 114 istypically a high capacity storage device for storing the large amountsof measurement data that can be obtained from the measurement sub-system122 during a data collection session. The data storage 114 may be calledupon to store data from several data collection sessions.

The user interface 116 broadly covers various mechanisms for user inputand output, and collectively refers to any combination of suitable userinput devices and output or display devices. The “user” can be a humanuser, or a machine user. In the case of a machine user, the interface116 can any suitable analog or digital communication interface forcommunication with another computing machine that is configured tooperate the gas analysis system of FIG. 1 by interfacing with the dataand control sub-system 124. In the case of a human user, the interface116 can include input devices such as a mouse pointing device, akeyboard, a graphics tablet, a touch screen device, and so on. Theinterface 116 can further include output devices such as a video displaymonitor, a simple set of LED indicators, a printing device, a removableflash memory device (e.g., USB thumb-drive), and so on. The interfaceallows the user to control the data and control sub-system 124 toconfigure the measurement sub-system to 122 to collect measurements andto store the measurements in the data storage 114.

The data and control sub-system 124 can be configured to any level ofsophistication as needed for a particular implementation of the gasanalysis system of FIG. 1, and typically is built to survive ruggedfield deployments for months or even years on end. The data and controlsub-system 124 can be a simple data logging component configured tocommunicate with the measurement sub-system 122 to simply receive datato be stored in the data storage 114. Examples of such a data andcontrol sub-system 124 include devices variously referred to in theindustry as data loggers; as measurement and control units,microloggers, and so on. Such devices can measure the instruments at aspecific scan rate, process data, and store the data.

In accordance with the present invention, the data and controlsub-system 124 can be a more full-featured and sophisticated datalogging component that is not only able to communicate with themeasurement sub-system 122 and receive measurement data from themeasurement sub-system to be stored in the data storage 114, but alsoincludes computer program code in accordance with the present inventionto produce temperature and pressure corrected gas flux values of thetarget gas of interest. It is noted that the data and control sub-system124 need not be near the measurement sub-system. In fact, depending onthe particular usage scenario of the present invention, the twosub-systems 122, 124 can be quite distant from each other.

Referring now to FIG. 2, the data processing component 112 of the dataand control sub-system 124 comprises a conventional data processor 202,for example, a central processing unit or a microcontroller or acombination of both, connected to other constituent components via thedata and control bus lines 222. The data processor 202 can be, orincludes, a programmable logic device (PLD) or a field programmable gatearray (FPGA), or other similar and commonly known logic devices. Thoseof ordinary skill in the art will appreciate that the data processingcomponent 112 may include many more components than those shown in FIG.2. However, it is not necessary that all of these generally conventionalcomponents be shown in order to disclose an illustrative embodiment forpracticing the present invention.

The data processing component 112 includes memory components 204, 206.The static memory component 206 is typically used to store the variouscomputer programs (e.g., operating system, applications) that areexecuted by the data processor 202. Typical static memories includeprogrammable read-only memory (PROM), flash RAM, and so on. Computerprogram instructions (or computer code) in accordance with the presentinvention (explained with respect to FIGS. 3 and 4 below) can be storedin the static memory 207. One of ordinary skill in the art willappreciate that when the computer program instructions are executed bythe data processor 202, different portions of the computer programinstructions may be loaded into the dynamic memory 204; e.g., randomaccess memory (RAM). One of ordinary skill will further appreciate thatcomputer program instructions can be stored in the data storage device114. The individual memories 204, 206 and the data storage 114illustrate various forms of computer-readable storage media. Dependingon the particular configuration, it will be appreciated that thecomputer program instructions according to the present invention can bestored for execution by the data processor 202 entirely on one ofmemories 204, 206 or data storage 114, or distributed among the memories204, 206 and data storage 114.

The data processing component 112 further includes a communicationinterface 208 which provides the circuitry and logic, includingelectrical and/or optical components, for communications with themeasurement sub-system 122. The communication interface 208 alsoincludes circuitry and logic for connecting to the data storage 114 andthe user interface 116. The communication interface 208 includes analogand digital circuitry to perform data acquisition in order to collectdata from the suite sensors that constitute the measurement sub-system122. In a particular embodiment, such data acquisition circuitry can beembodied in a separate component commonly referred to as a dataacquisition card.

Referring now to FIGS. 3 and 4, a discussion of the computer programcode in accordance with the present invention for computing gas flux fora target gas that is corrected for temperature and pressure conditions.The program code will be explained in terms of the flowchartsillustrated in the figures rather than in terms of actual lines ofprogram code, since the later would include details not relevant to thediscussion of the invention. The flowcharts in FIGS. 3 and 4 are almostthe same, with the exception of measurements for water vapor in FIG. 4.FIG. 3 illustrates a situation where water content in the target gas isnot significant, while FIG. 4 covers the more general situation wherewater content in the target gas is considered.

The computer program code is typically executed by the data processor202 (e.g., a CPU or a microcontroller). Typically, the computer programcode includes computer instructions, also known as machine language,that cause the data processor to perform various control operations anddata manipulation operations. In addition, to traditional machine codelanguage that is executed by physical central processing units (e.g.,Intel processors, Texas Instruments microcontrollers, and the like), thecomputer instructions can be provided for so-called “virtual machines.”A popular example is the Java programming language which can be compiledto produce “bytecode” that runs on a Java virtual machine (JVM). Thisand other alternatives to traditional CPU architectures can be readilyaccommodated in accordance with the present invention. The flowcharts inFIGS. 3 and 4 represent high-level processing steps in accordance withthe disclosures of the present invention. One of ordinary skill in theart will readily appreciate that the flows (e.g., FIGS. 3 and 4) can beimplemented for a specific data processor or data processors in myriadways, including but not limited to the data processor 202. For example,if the data processor 202 includes one or more PLD, FPGA, or some othersimilar kind of logic device, then it will be appreciated by those ofordinary skill in the that some or all of the processing (e.g., FIGS. 3and 4) may be implemented in such logic device(s).

Referring to the particular flow shown in FIG. 3, the process beginswith data collection steps 302-308. In step 302, the gas analyzer 102operates to perform a series of measurements of gas density of thetarget gas. A series of gas density measurement data are produced by thegas analyzer 102, and obtained by the data processor 202 to be stored indata storage 114. In step 304, the wind speed measuring device 104operates to perform a series of measurements to measure the wind speedduring the same time that the gas density measurements are beingperformed. The series of wind speed measurement data that is produced isobtained by the data processor 202 and stored in data storage 114. Instep 306, a series of measurements are performed by the temperaturesensor 108 to obtain temperature measurement data which are then storedin data storage 114.

The data collection rate is typically on the order of 10 Hz or so; i.e.,10 Hz means 10 samples are collected per second. The sampling rate canbe higher or lower as ambient conditions require and/or depending on themeasurement devices. For example, in order to compute gas flux using theEddy Covariance technique (or using other similar techniques), theinstrument needs to sample all relevant air parcels traveling up anddown. If the instrument is too slow, then it will miss flux transport insmall and fast movements. Typically, people collect data at rates of10-20 times per second. An advantageous aspect of the present inventionis that the low end of the range of data collection rates can be verylow (e.g., 5 Hz) while still allowing correct flux determinations. Thus,for closed path or open path systems with very tall towers, the datacollection rate can be 5 times per second and still measure fluxcorrectly because the higher you go the bigger and slower the motionsare involved in most of the air and gas transport. When people measurewith airplanes, they may go to 40 Hz or more due to fast travellingthrough these motions.

Typically, the data collection rate for each measurement (gas density,wind speed, temperature) is the same. Thus, for a data collection rateof 10 Hz that means ten gas density measurements are taken per secondand stored, and ten wind speed measurements and ten temperaturemeasurements are taken per second and stored. However, it is possible tocollect the different kinds of measurements at different rates. Forexample, the gas density measurements can be collected at say 30 Hz,while the wind speed measurements are collected at 40 Hz and thetemperature measurements are collected at 50 Hz. However, the foregoinganalytical techniques require that the measurements be correlated withwind speed. Thus, using conventionally known digital data processingtechniques, such sub-sampling or averaging, the wind speed can besub-sampled into 30 Hz data in order for the gas density data to becorrelated with the wind speed data. Likewise, the temperature data canbe sub-sampled down to 40 Hz data in order for the temperature data tobe properly correlated with the wind speed data.

In step 308, a sensible heat flux value is obtained. This particularembodiment of the present invention is suitable for ambient conditionsthat are sufficiently dry so as not to require taking water vapormeasurements. Since sensible heat flux is typically computed based onwater vapor measurements, for this particular embodiment, where thereare no water vapor measurements, the sensible heat flux can be obtainedby taking measurements using a scintillometer or a LIDAR, or canestimated from known and conventional modeling techniques using solarradiation, soil moisture, etc.

Steps 302-308 in FIG. 3 relate to collecting, modeling (in the case ofsensible heat flux), or otherwise obtaining some of the measurementsused in the analytical techniques of the present invention. Theremainder of FIG. 3 relates to the analytical techniques to determine acorrected gas flux of the target gas. Reference will be made to theequations and derivations discussed above.

In step 310, a “raw” uncorrected flux value w′ρ_(cm)′ is computed fromthe gas density measurements obtained in step 302. The raw flux is avalue computed based on the measured data, as discussed in connectionwith Equation 11 above. As explained, the raw flux is w′ρ_(cm)′, wherew′ is the deviation of a wind speed measurement from the mean (average)value of all of the vertical wind speed measurements. The term w′represents the average of all such deviations. Similarly, ρ_(cm)′ is thedeviation of a gas density measurement of the target gas (e.g., methane)from the mean (average) value of all of the gas density measurements,and the term ρ_(cm′) represents the average of all such deviations. Theparticular mathematical expression for the uncorrected flux will varydepending on the particular method of calculating the flux.

In step 312, a thermal expansion term is computed from the measurementsobtained in steps 302-308. The thermal expansion term

$\frac{S}{\overset{\_}{\rho \;}C_{p}}\frac{\overset{\_}{\rho_{cm}}}{\overset{\_}{T}}$

represents the effect of thermal expansion effect on the measured fluxof the target gas in connection with the WPL formulation for fluxcalculations. The terms are explained above. It will be appreciated byone of ordinary skill in the art that this thermal expansion term is notunique to the WPL formulation, but rather is a term that represents thethermal expansion effects on gas flux due to heat flux, and is a termthat is present in most models. The particular mathematical expressionof the thermal expansion term, however, will vary depending on theparameters of the specific model.

In step 314, correction factors are determined in accordance with thepresent invention as discussed above in connection with the equationsabove. In particular, Equation 11 represents the most general form andaccounts for all temperature and pressure effects on gas absorptionlines upon which gas density measurements are made. Equation 30, showsan example of compensating for the effect of temperature and pressure onthe measurement response of a specific instrument; e.g., the LI-7700 gasanalyzer of the assignee of the present invention. Equation 36 is aspecific instance of Equation 30, determined for a given pressure valueand a range of temperatures. Since the discussion of FIG. 3 assumes theeffect of water dilution is negligible and thus being ignored, the waterdilution term in Equations 11, 30, and 36, namely

${\mu \frac{E}{\overset{\_}{\rho_{d}}}\frac{\overset{\_}{\rho_{cm}}}{1 + {\mu \frac{\overset{\_}{\rho_{v}}}{\overset{\_}{\rho_{d}}}}}},$

can be dropped out of the equations.

An example of the correction factors is shown in Equation 36 as thecoefficients of the terms w′ρ_(cm)′ and

$\frac{S}{\overset{\_}{\rho \;}C_{p}}\frac{\overset{\_}{\rho_{cm}}}{\overset{\_}{T}}$

from steps 310 and 312, namely 0.99 and 1.31. These coefficients wereobtained by considering the response function of the specific instrumentused to measure gas density, and in particular the response function atthe same pressure as when gas density was measured. A discussion of thedetermination of the specific coefficient values shown in Equation 36 isexplained in connection with Equations 31-35 above.

Typically, the response function is calculated using specificspectroscopic parameters from the HITRAN database and experimentallyvalidated by taking a series of gas density measurements over a range oftemperatures at a given density of the target gas at a given pressure.The HITRAN (High Resolution TRANsmission Molecular Absorption Database)database is a recognized international standard publicly availablecompilation of spectroscopic reference data. Of course, sources forspectroscopic reference data other than the HITRAN database cancertainly be used. In addition, as another source of spectral data,there are a variety of open source software, web based simulations, andthe like that allow a person to run any simulations using a complete setof parameters. Also, it will be appreciated that any of a number ofknown techniques for calibrating an instrument can be used.

FIG. 5A illustrates an example of the result of such calculations fittedwith a linear approximation to obtain the calibration curve shown in thefigure. FIG. 5B shows the same data fitted with an exponential model.This data was calculated at a given pressure; e.g. 97 kPa (kilopascals)over a temperature range from 233K to 323K (kelvin). The horizontal axisis a normalized temperature scale where the temperature range isnormalized by the mean temperature ( T, T-overbar). The vertical axis isa value of the χ function at a specific temperature and pressure.

A calibration “surface” (response surface) is obtained if one obtainsseveral calibration curves by repeating the foregoing calculationsor/and data collection for several values of pressure. The calibrationsurface then can be used to select an appropriate calibration curvebased on the pressure under which the measurements (steps 302-308) wereobtained. The selected calibration curve is then used to determine thecoefficients for Equation 36, as explained in connection with Equations31-35, or alternatively in connection with Equations 37-39.

In step 316, the terms obtained in steps 310 and 312 are adjusted by thecorrection factors obtained in step 314. In the particular embodimentper Equation 36 of the present invention, the coefficients 0.99 and 1.31are multiplied respectively with the terms w′ρ_(cm)′; and

$\frac{S}{\overset{\_}{\rho \;}C_{p}}{\frac{\overset{\_}{\rho_{cm}}}{\overset{\_}{T}}.}$

The corrected flux F_(c) is then computed (step 318) in accordance withEquation 36 by summing the adjusted terms, specifically in the casewhere water dilution is not a significant effect and can be ignored, thecorrected flux is:

$F_{c} = {{0.99\overset{\_}{w^{\prime}\rho_{cm}^{\prime}}} + {1.31\frac{S}{\overset{\_}{\rho}\; C_{p}}{\frac{\overset{\_}{\rho_{cm}}}{\overset{\_}{T}}.}}}$

Referring now to FIG. 4, recall from the introductory discussion aboveof FIGS. 3 and 4, that unlike FIG. 3, FIG. 4 represents a scenario wherethe effects of water content in the target gas cannot be ignored. Thesteps in FIG. 4 common to those in FIG. 3 are identified with the samereference numerals and their discussion need not be repeated here. InFIG. 4, there is an additional measurement, namely the water vapormeasurements, step 402. In step 404, a water dilution term is computedfrom the measurements obtained in steps 302-308, 402. The water dilutionterm

$\mu \frac{E}{\overset{\_}{\rho_{d}}}\frac{\overset{\_}{\rho_{cm}}}{1 + {\mu \frac{\overset{\_}{\rho_{v}}}{\overset{\_}{\rho_{d}}}}}$

represents latent heat flux effect (due to the presence of water vaporin the target gas) on the measured flux of the target gas in connectionwith the WPL formulation for flux calculations. It will be appreciatedby one of ordinary skill in the art that the above water dilution termis not unique to the WPL formulation, but rather is a term thatrepresents the effect on the target gas flux due to water content in thetarget gas that is modeled by most models. The particular mathematicalexpression of the water dilution term, however, will vary depending onthe parameters of the specific flux model.

In step 406, the correction factors are computed as explained above inFIG. 3 for step 314. Since water vapor effects are being considered, thediscussion of Equations 34-36 include the effects of water dilution,namely the term

$\mu \frac{E}{\overset{\_}{\rho_{d}}}{\frac{\overset{\_}{\rho_{cm}}}{1 + {\mu \frac{\overset{\_}{\rho_{v}}}{\overset{\_}{\rho_{d}}}}}.}$

In step 408, the terms obtained in steps 310, 312, and 408 are adjustedby the correction factors obtained in step 406. In the particularembodiment per Equation 36 of the present invention, the coefficients0.99, 099, and 1.31 are multiplied respectively with the termsw′ρ_(cm)′,

${\mu \frac{E}{\overset{\_}{\rho_{d}}}\frac{\overset{\_}{\rho_{cm}}}{1 + {\mu \frac{\overset{\_}{\rho_{v}}}{\overset{\_}{\rho_{d}}}}}},\mspace{14mu} {{and}\mspace{14mu} \frac{S}{\overset{\_}{\rho \;}C_{p}}{\frac{\overset{\_}{\rho_{cm}}}{\overset{\_}{T}}.}}$

The corrected flux F_(c) is then computed (step 410) in accordance withEquation 36 by summing the adjusted terms.

One of ordinary skill in the art will appreciate that many variations ofthe foregoing teachings of the present invention are possible. One ofordinary skill will appreciate that the particular analyticalderivations disclosed herein will vary due to differences betweenparticular gas analyzers, but that such variations can be readilyaccommodated in view of the teachings of the present invention. Those ofordinary skill in the relevant arts will appreciate that any of a numberof alternate embodiments of the present invention are well within theirskill, and the present invention can therefore be practiced withoutundue experimentation.

1. A computer-readable storage medium having stored thereon computerprogram code for determining gas flux of a target gas, the computerprogram code effective for being executed by a data processor, thecomputer program code comprising: computer instructions configured tocause the data processor to receive a plurality of gas densitymeasurement data for the target gas, a plurality of wind speedmeasurement data, and a plurality of temperature measurement data;computer instructions configured to cause the data processor todetermine a raw flux term of the target gas based on the gas densitymeasurement data for the target gas and the wind speed measurement data;computer instructions configured to cause the data processor to obtain aheat flux measurement indicative of heat flux during a period of timewhen the gas density measurement data was obtained; computerinstructions configured to cause the data processor to determine athermal expansion term based on the sensible heat flux measurement andthe temperature measurement data; and computer instructions configuredto cause the data processor to determine at least a first multiplicationfactor based on a response function of the instrument used to obtain thegas density measurement data, wherein the response function of theinstrument relates actual gas density and measured gas density of thetarget gas as a function of temperature, wherein a gas flux value of thetarget gas is generated by combining one of the raw flux term or thethermal expansion term with said at least a first multiplication factor.2. The computer-readable storage medium of claim 1 wherein the responsefunction is calibration data of the instrument determined for anoperating pressure substantially equal to the pressure at which the gasdensity measurement data was obtained.
 3. The computer-readable storagemedium of claim 1 further comprising computer instructions configured tocause the data processor to determine a second multiplication factorbased on the response function of the instrument, wherein the gas fluxvalue of the target gas is generated by combining the raw flux term withsaid at least a first multiplication factor and combining the thermalexpansion term with the second multiplication factor.
 4. Thecomputer-readable storage medium of claim 1 wherein the gas measurementdata of the target gas is obtained by sensing the target gas flowingacross a sampling path, wherein the temperature measurement data isobtained at a location proximate the sampling path.
 5. Thecomputer-readable storage medium of claim 1 wherein the gas measurementdata of the target gas is obtained by sensing the target gas flowingacross a sampling path, wherein the temperature measurement data isobtained in the sampling path.
 6. The computer-readable storage mediumof claim 1 wherein the wind speed measurement data is vertical windspeed measurement data.
 7. The computer-readable storage medium of claim1 wherein the sensible heat flux is obtained by taking measurements withone of a sonic anemometer, a fine-wire thermocouple, or ascintillometer.
 8. The computer-readable storage medium of claim 1wherein the sensible heat flux is based on the temperature measurementdata.
 9. A computer-readable storage medium having stored thereoncomputer program code for determining gas flux of a target gas, thecomputer program code effective for being executed by a data processor,the computer program code comprising: computer instructions configuredto cause the data processor to receive a plurality data, the pluralityof data including: a plurality of gas density measurement data for thetarget gas; a plurality of wind speed measurement data indicative ofspeed of movement of the target gas; and a plurality of temperaturemeasurement data; computer instructions configured to cause the dataprocessor to determine a raw flux of the target gas based on the gasdensity measurement data for the target gas and the wind speedmeasurement data; computer instructions configured to cause the dataprocessor to determine a thermal expansion term based on the temperaturemeasurement data; and computer instructions configured to cause the dataprocessor to compute the gas flux of the target gas based on the rawflux and the thermal expansion term, at least one of the raw flux or thethermal expansion term being adjusted by a multiplication factordetermined based on an instrument response function of the instrumentused to obtain the gas density measurement data, the response functionrelating actual gas density and measured gas density of the target gasas a function of temperature.
 10. The computer-readable storage mediumof claim 9 wherein the raw flux and the thermal expansion term each isadjusted by a respective multiplication factor determined based on theinstrument response function.
 11. The computer-readable storage mediumof claim 9 wherein the plurality of data further includes a plurality ofwater vapor density measurement data indicative of water content in thetarget gas, the computer program further comprising computerinstructions configured to cause the data processor to determine a waterdilution effect based on the water vapor density measurement data,wherein the gas flux of the target gas is further based on the waterdilution effect adjusted by a respective multiplication factordetermined based on the instrument response function.
 12. Thecomputer-readable storage medium of claim 11 wherein the plurality ofdata further includes dry air density measurement data, wherein thewater dilution effect is further based on the dry air densitymeasurement data.
 13. The computer-readable storage medium of claim 9wherein the response function is calibration data of the instrumentdetermined for an operating pressure substantially equal to the pressureat which the gas density measurement data was obtained.
 14. Thecomputer-readable storage medium of claim 9 wherein the gas measurementdata of the target gas is obtained by sensing the target gas in asampling path, wherein the temperature measurement data is obtained at alocation proximate the sampling path.
 15. A gas analyzer systemcomprising: a gas analyzer having an optical path and operable toproduce a plurality of gas density measurements when a target gas flowsacross the optical path; a wind speed detector disposed in proximity tothe gas analyzer; a temperature sensor disposed in proximity to the gasanalyzer and clear of the optical path of the gas analyzer; and acontroller configured to: receive a plurality of gas density measurementdata obtained by the gas analyzer; receive a plurality of wind speedmeasurement data obtained by the wind speed detector; receive aplurality of temperature measurement data obtained by the temperaturesensor; determine a raw flux term of the target gas based on the gasdensity measurement data and the wind speed measurement data; determinea thermal expansion term based on the temperature measurement data; andcompute the gas flux of the target gas based on the raw flux term andthe thermal expansion term, wherein at least one term being adjusted bya multiplication factor determined based on an instrument responsefunction corresponding to the instrument used to obtain the gas densitymeasurement data, the instrument response function relating actual gasdensity and measured gas density of the target gas as a function oftemperature.
 16. The system of claim 15 wherein each of the raw fluxterm and the thermal expansion term is adjusted by a respectivemultiplication factor determined based on the instrument responsefunction.
 17. The system of claim 15 wherein the instrument responsefunction is calibration data of the instrument determined for anoperating pressure substantially equal to the pressure at which the gasdensity measurement data was obtained.
 18. The system of claim 15further comprising a water vapor detector to detect water content in thetarget gas, the controller further configured to receive a plurality ofwater vapor density measurement data, wherein the gas flux of the targetgas is further based on the water dilution term adjusted by a respectivemultiplication factor determined based on the instrument responsefunction.
 19. The system of claim 15 wherein the controller is in datacommunication with the gas analyzer, the wind speed detector, and thetemperature sensor.
 20. The system of claim 15 further comprising acommunication module in data communication with the gas analyzer, thewind speed detector, and the temperature sensor to store data receivedtherefrom, the controller being in communication with the communicationmodule to receive data stored therein.
 21. A gas analysis data handlingdevice comprising: a processor; a communication interface to receivedata from one or more measuring devices; and a memory store for storingdata received by the communication interface, the processor configuredto: receive to receive a plurality data, the plurality of dataincluding: a plurality of gas density measurement data for the targetgas; a plurality of wind speed measurement data indicative of speed ofmovement of the target gas; and a plurality of temperature measurementdata; determine a raw flux of the target gas based on the gas densitymeasurement data for the target gas and the wind speed measurement data;determine a thermal expansion effect term based on the temperaturemeasurement data; and compute the gas flux of the target gas based onthe raw flux and the thermal expansion effect, at least one of the rawflux or the thermal expansion effect term being adjusted by amultiplication factor determined based on an instrument responsefunction of the instrument used to obtain the gas density measurementdata, the response function relating actual gas density and measured gasdensity of the target gas as a function of temperature.
 22. The deviceof claim 21 wherein each of the raw flux and the thermal expansioneffect term is adjusted by a respective multiplication factor determinedbased on the instrument response function.
 23. The device of claim 21wherein the response function is calibration data of the instrumentdetermined for an operating pressure substantially equal to the pressureat which the gas density measurement data was obtained.
 24. The deviceof claim 21 further comprising a computer-readable memory connected tothe processor, the computer-readable memory having stored thereinprogram code, the program code comprising: computer instructionsconfigured to cause the processor to receive the plurality data;computer instructions configured to cause the processor to determine theraw flux of the target gas based on the gas density measurement data forthe target gas and the wind speed measurement data; computerinstructions configured to cause the processor to determine the thermalexpansion effect term based on the temperature measurement data; andcomputer instructions configured to cause the processor to compute thegas flux of the target gas based on the raw flux and the thermalexpansion effect, at least one of the raw flux or the thermal expansioneffect term being adjusted by the multiplication factor.
 25. A gasanalysis data handling device comprising: a processor; a communicationinterface to receive data from one or more measuring devices; and amemory store for storing data received by the communication interface,the processor configured to: receive a plurality data, the plurality ofdata including: a plurality of gas density measurement data for thetarget gas; a plurality of wind speed measurement data indicative ofspeed of movement of the target gas; a plurality of water vapor densitymeasurement data indicative of water content in the target gas; and aplurality of temperature measurement data; determine a raw flux of thetarget gas based on the gas density measurement data for the target gasand the wind speed measurement data; determine a water dilution effectbased on the water vapor density measurement data; determine a thermalexpansion value based on the temperature measurement data; and computethe gas flux of the target gas based on the raw flux, the water dilutioneffect, and the thermal expansion value, wherein at least one of the rawflux, the water dilution effect, or the thermal expansion value isadjusted by a multiplication factor determined based on an instrumentresponse function of the instrument used to obtain the gas densitymeasurement data, the response function relating actual gas density andmeasured gas density of the target gas as a function of temperature. 26.The device of claim 25 wherein each of the raw flux, the water dilutioneffect, and the thermal expansion value is adjusted by a respectivemultiplication factor determined based on an instrument responsefunction of the instrument used to obtain the gas density measurementdata, the response function relating actual gas density and measured gasdensity of the target gas as a function of temperature
 27. The device ofclaim 25 wherein the response function is calibration data of theinstrument determined for an operating pressure substantially equal tothe pressure at which the gas density measurement data was obtained. 28.The device of claim 25 further comprising a computer-readable memoryconnected to the processor, the computer-readable memory having storedtherein program code, the program code comprising: computer instructionsconfigured to cause the processor to receive the plurality data;computer instructions configured to cause the processor to determine theraw flux of the target gas based on the gas density measurement data forthe target gas and the wind speed measurement data; computerinstructions configured to cause the processor to determine the waterdilution effect based on the water vapor density measurement data;computer instructions configured to cause the processor to determine thethermal expansion value based on the temperature measurement data; andcomputer instructions configured to cause the processor to compute thegas flux of the target gas based on the raw flux, the water dilutioneffect, and the thermal expansion value, wherein at least one of the rawflux, the water dilution effect, or the thermal expansion value isadjusted by the multiplication factor.